If $\sum{a_n}=x$ and $\sum{n a_n}=y$ and both converge, is there any relation between $x$ and $y$?
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5No. e.g. consider the sequence $(a_0, a_1, a_2, \ldots ) = (x-y,y,0,\ldots)$. – achille hui Aug 06 '14 at 10:25
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3If the sums start at $n=1$ and the entries $a_n$ are nonnegative then $x\leqslant y$. Otherwise, see @achille's comment. – Did Aug 06 '14 at 10:51
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I asked because I noticed that $\zeta(3)=\sum{1/n^3}=$Apéry's constant and $\zeta(2)=\sum{1/n^2}=\sum{n/n^3}=\frac{\pi^2}{6}$. Given the answer here I asked another question. – Mark Hurd Aug 06 '14 at 12:14
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if $\lim_{n \to \infty} \sqrt[n]{|a_n|} \lt 1$ then the function: $$ f(z) = \sum_{k=0}^{\infty} a_nz^n $$ has a radius of convergence $\gt1$. consequently it is holomorphic on the unit disc. its derivative $\frac{df}{dz} = g(z)$ has the same radius of convergence as $f(z)$ and: $$ f(1) = x \\ g(1) = y $$
David Holden
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